Question

Find a natural number whose square diminished by 84 is equal to thrice of 8 more than the given number.

Answer 1

Let the given number be x.
Given that the square of a natural number diminished by 84 is equal to thrice of 8 more than the given number.
$$\begin{gathered} \Rightarrow x^2-84=3\left(8+x\right) \\ \Rightarrow x^2-84=24+3x \\ \Rightarrow x^2-3x-108=0 \\ \Rightarrow x^2-12x+9x-108=0 \\ \Rightarrow x(x-12)+9(x-12)=0 \\ \Rightarrow (x+9)(x-12)=0 \\ \Rightarrow x=-9\mathrm{or}x=12 \end{gathered}$$
We ignore the negative value as we have taken natural numbers under consideration.
Hence, x = 12 is the required number.